Alternating factorial

In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials of positive integers.

This is the same as their sum, with the odd-indexed factorials multiplied by −1 if n is even, and the even-indexed factorials multiplied by &minus;1 if n is odd, resulting in an alternation of signs of the summands (or alternation of addition and subtraction operators, if preferred). To put it algebraically,


 * $$\operatorname{af}(n) = \sum_{i = 1}^n (-1)^{n - i}i!$$

or with the recurrence relation


 * $$\operatorname{af}(n) = n! - \operatorname{af}(n - 1)$$

in which af(1) = 1.

The first few alternating factorials are


 * 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019

For example, the third alternating factorial is 1! – 2! + 3!. The fourth alternating factorial is −1! + 2! − 3! + 4! = 19. Regardless of the parity of n, the last (nth) summand, n!, is given a positive sign, the (n – 1)th summand is given a negative sign, and the signs of the lower-indexed summands are alternated accordingly.

This pattern of alternation ensures the resulting sums are all positive integers. Changing the rule so that either the odd- or even-indexed summands are given negative signs (regardless of the parity of n) changes the signs of the resulting sums but not their absolute values.

proved that there are only a finite number of alternating factorials that are also prime numbers, since 3612703 divides af(3612702) and therefore divides af(n) for all n &ge; 3612702. The primes are af(n) for
 * n = 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, ...

with several higher probable primes that have not been proven prime.